where \({\rm {exp}}_{x_0;h} (V)\)  denotes the time 1 map of \(h\) -geodesics with initial point \(x_0\)  and initial velocity \(V\in T_{x_0}\partial M\) . The coordinate \(t^{\prime }\in \rho \mapsto x((t^{\prime },0); x_0)\)  is then an \(h\) -geodesic coordinate system for a neighborhood of \(x_0\)  on the boundary surface \(\partial M\) . We can extend this to become a coordinate system for points in \(M\)  near \(x_0\)  so that \(t \mapsto x(t;x_0)\)  is a boundary normal coordinate system with \(t_3 >0\)  in \(M\)  as the boundary defining function. Readers wishing to know more about boundary normal coordinates can refer to [1]} for a brief recollection of the basic properties we use here and Prop 5.26 of [2]} for a detailed construction.
